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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>;<br/><b>let </b><font color="Maroon" title="c4">F1</font> be  <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>;<br/><b>let </b><font color="Maroon" title="c5">G1</font>, <font color="Maroon" title="c6">G2</font> be  <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>;<br/><b>let </b><font color="Maroon" title="c7">f</font> be    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font>;<br/>













<b>assume </b><a NAME="E1:54"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> 
 ;<br/>

<a NAME="E2:54"/><b>then </b><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c5">G1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">G2</font>
 
<b>by </b><i><a class="ref" href="functor3.html#D4" target="_self" title="FUNCTOR3:def.4">Def4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">k</font> = <font color="Maroon" title="c7">f</font> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c4">F1</font> as    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c5">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> <b>by </b><i><a class="ref" href="functor3.html#T29" target="_self" title="FUNCTOR3:th.29">Th29</a></i>;<br/>
<div><i><font color="Green" title="E33">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>;<br/><a NAME="E1:54_1"/><i><font color="Green" title="E34">A4</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span> )
 <b>by </b><i><a class="ref" href="functor0.html#T34" title="FUNCTOR0:th.34">FUNCTOR0:34</a></i>;<br/><a NAME="E2:54_1"/>
<font color="Maroon" title="c5">G1</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c6">G2</font>
 <b>by </b><i><a class="txt" href="functor3.html#E1:54"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="functor3.html#D4" target="_self" title="FUNCTOR3:def.4">Def4</a></i>;<br/><a NAME="E3:54_1"/><b>then </b>
<font color="Maroon" title="c8">k</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="functor3.html#T12" target="_self" title="FUNCTOR3:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E4:54_1"/>
<font color="Maroon" title="c8">k</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 <b>by </b><i><a class="txt" href="functor3.html#E1:54"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="functor3.html#E1:54_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="functor3.html#D5" target="_self" title="FUNCTOR3:def.5">Def5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:54"/>
<font color="Maroon" title="c5">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </b></a><div class="add">

<a NAME="E1:54_2"/>
<font color="Maroon" title="c4">F1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c4">F1</font>
 
<b>by </b><i><a class="ref" href="functor2.html#T9" title="FUNCTOR2:th.9">FUNCTOR2:9</a></i>;<br/>
<b>hence </b><a NAME="E2:54_2"/>
<font color="Maroon" title="c5">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>
 <b>by </b><i><a class="txt" href="functor3.html#E2:54"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="functor3.html#E32"><i><font color="Green" title="E21">Lm2</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor3.html#D4" target="_self" title="FUNCTOR3:def.4">FUNCTOR3:def 4</a><br/>

<a NAME="E3:54_2"/>
<font color="Maroon" title="c6">G2</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c5">G1</font>
 
<b>by </b><i><a class="txt" href="functor3.html#E1:54"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="functor3.html#D4" target="_self" title="FUNCTOR3:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E4:54_2"/>
<font color="Maroon" title="c6">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c5">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>
 <b>by </b><i><a class="ref" href="functor3.html#T10" target="_self" title="FUNCTOR3:th.10">Th10</a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c8">k</font>
;<br/>

<b>let </b><font color="Maroon" title="c9">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>;<br/>

<b>thus </b><a NAME="E5:54_2"/>
<font color="Maroon" title="c8">k</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 <b>by </b><i><a class="txt" href="functor3.html#E4:54"><i><font color="Green" title="E33">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:54"/>
( <font color="Maroon" title="c5">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c7">f</font> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c4">F1</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c5">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> )
 <b>by </b><i><a class="txt" href="functor3.html#E4:54"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="functor3.html#D5" target="_self" title="FUNCTOR3:def.5">Def5</a></i>;<br/>


</div>
